This ia a great question and you have insightful observations about it. I agree with your coclusion that the problem here is to decide how to set off the texting. I would not use underlining because that adds an entity not present in the print and while underlining is a new format, I think it works best with single words or phrases and not with protracted passages such as this. Imagine what underlining all that would loook like in print!
I suggest boxes. Boxing is the most obvious and essential way to set off large print selections in braille. Use conventional capitalization and I think 1-3 dialog even though there are no speaker names. Don't the angle brackets indicate that the narrator is texting?
Write a TN that says the text messageing is printed in uppercase letters and that in braille the text message exchanges are enclsosed in boxes and the updercase print is omitted. I would keep the angle brackets as printed without comment and that reading experience will remain the same for the braille reader as it is for the print reader. List the angle brackets as special symbols.